Actually, I expected they'd do exactly that, and was disappointed when they didn't. It's basically the same reason why Sun created Java. They made Java so that software developers could make software that ran on Windows and Linux. Oh eh, and on Sun computers. Which were very much losing the server OS wars at the time. You might say that Java didn't save Sun, but I like to think that maybe instead Java postponed its demise by many years.
I think the same could've happened with Nokia. If Nokia sponsored the best way to make fast, snappy, great apps, in little time, that work comfortably and with a native feel on iOS and Android (oh eh, and also Nokia's current OS hobby), people'd churn out Nokia versions of their apps without much extra work. This might've significantly increased the app ecosystem for Nokia phones, which might have made more people buy the phones. A lot of people (and very much not just the geeks) do care that their favourite apps are supported and decent on a particular handset.
I never understood why they didn't do this. At the time when many developers were still a bit nervous about learning Objective-C just to support one single platform, a good, company-supported Qt-for-all-phones could've made Qt the leading app toolkit by far. Qt sure had the head start, Nokia sure had the resources, and the Qt team sure had the skills. Sad.
Nokia bought Trolltech in early-to-mid 2008 (initial offer announced in January, purchase finalized in June). The iPhone SDK had been announced in October 2007 and been made available in February, Android 1.0 would only be released in September (though the beta had started in November 2007), the Windows 7 RTM would only happen in 2009 and W7P in 2010 so the Windows world was still "old windows" (on which regular Qt worked) and Windows Mobile 6.
One can easily assume that, as soon as the deal was announced and likely to go through, Trolltech had little reason to work on porting Qt to non-nokia platforms (out of the 2 months or so they'd have had for the Android beta, and negative 2 months for iOS).
And once Trolltech/Qt was part of Nokia, why would Nokia fund Qt/Android or Qt/iOS? The original plan was to build Nokia's own "next-generation platform" Maemo with it after all.
As to third parties, they did attempt porting Qt (ultimately fizzling out, for the most part):
* qt-iphone (Qt/Cocoa Touch) http://www.qt-iphone.com/Introduction.html
* Necessitas (Qt/Android) http://sourceforge.net/p/necessitas/home/necessitas/